Maximal estimates for averages over degenerate hypersurfaces

TL;DR

This paper establishes optimal bounds for maximal averages over hypersurfaces with specific Fourier decay, resolving a conjecture by Stein and extending previous results to non-flat surfaces.

Contribution

It proves the optimal $L^p$ bounds for maximal averages over hypersurfaces with Fourier decay rate 1/2, confirming Stein's conjecture and generalizing earlier work.

Findings

Established optimal maximal bounds for hypersurfaces with Fourier decay 1/2.

Verified $L^p$ boundedness for non-flat hypersurfaces.

Resolved Stein's conjecture on maximal averages.

Abstract

We study $L^p$ boundedness of the maximal average over dilations of a smooth hypersurface $S$. When the decay rate of the Fourier transform of a measure on $S$ is $1/2$, we establish the optimal maximal bound, which settles the conjecture raised by Stein. Additionally, when $S$ is not flat, we verify that the maximal average is bounded on $L^p$ for some finite $p$, which generalizes the result by Sogge and Stein.